Fast least-squares reverse time migration via a superposition of Kronecker products

Abstract

Least-squares reverse time migration (LSRTM) has become increasingly popular for complex wavefield imaging due to its ability to equalize image amplitudes, attenuate migration artifacts, handle incomplete and noisy data, and improve spatial resolution. The major drawback of LSRTM is the considerable computational cost incurred by performing migration/demigration at each iteration of the optimization. To ameliorate the computational cost, we introduced a fast method to solve the LSRTM problem in the image domain. Our method is based on a new factorization that approximates the Hessian using a superposition of Kronecker products. The Kronecker factors are small matrices relative to the size of the Hessian. Crucially, the factorization is able to honor the characteristic block-band structure of the Hessian. We have developed a computationally efficient algorithm to estimate the Kronecker factors via low-rank matrix completion. The completion algorithm uses only a small percentage of preferentially sampled elements of the Hessian matrix. Element sampling requires computation of the source and receiver Green’s functions but avoids explicitly constructing the entire Hessian. Our Kronecker-based factorization leads to an imaging technique that we name Kronecker-LSRTM (KLSRTM). The iterative solution of the image-domain KLSRTM is fast because we replace computationally expensive migration/demigration operations with fast matrix multiplications involving small matrices. We first validate the efficacy of our method by explicitly computing the Hessian for a small problem. Subsequent 2D numerical tests compare LSRTM with KLSRTM for several benchmark models. We observe that KLSRTM achieves near-identical images to LSRTM at a significantly reduced computational cost (approximately 5–15× faster); however, KLSRTM has an increased, yet manageable, memory cost.